Review: Revenue Function, R(X); Cost Function, C(X); Profit Function, P(X)

Mac 2233Section 3.5 Marginal Analysis

Review: Revenue function, R(x); Cost function, C(x); Profit function, P(x)

P(x) = R(x) – C(x)

C(x) is the total cost of producing x items.

New Terms:

is called the marginal cost function. It estimates the cost of producing the next item. It is the rate of change of the cost at x.

is called the marginal revenue function. It estimates the increase in revenue that results when one more item is sold, if x represents the number of items sold. It is the rate of change of R at x.

is called the marginal profit function. It estimates the increase in profits that results when one more item is sold, if x represents the number of items sold. It is the rate of change of P at x.

is the average cost per item for x items produced. = This is an average in the same sense that averaging grades is an average. It is the total cost to produce x items divided by x, the total number of items. So this gives the average cost of one item when x items are produced.

Interpret the following in 1-5:

1. The approximate cost of producing the 101st item is $6.00. You could also say that at a production level of 100 items, the approximate rate of change in cost is $6/item.

2. ______

______

3. ______

______

4. The next item, the 101st item, sold will increase the revenue by approximately $20,000. You could also say that at a selling level of 100 items, the approximate rate of change in revenue is $20,000/item.

5. ______

______.

6. Given: C(x) = 150,000 + 20x – 0.0001x2 (Note fixed costs are $150,000)

a. Find the marginal cost function, ______

b. Use the marginal cost function to estimate the cost of the 50,001st item.______

(Put unit of measurement.)

c. Give the average cost function and find the average cost when 50,000 items are produced.

d. Is the average cost increasing or decreasing at this productions level as more items are produced?

7. Find the marginal cost at the given production level and state the units of measurement.

C(x) = 30,000 + 4x - , x = 30,000.

First find the marginal cost function:______

Then substitute 30,000 for x in the marginal cost function.______

8. Refer to the cost function in problem 7 above.

a. Give the approximate cost of producing the 30,001 st item.______

b. Give the total cost of producing 30,000 items.______

c. Give the average cost function.______

d. Give the average cost per item when 30,000 items are produced.______

9. Given that C(x) = -0.01x2 + 2x + 100 and that R(x) = x2 + 16x + 24. Find the

(a) marginal cost function.______

(b) marginal revenue function.______

10. Using the functions in 9 above, evaluate

a. and interpret.______

b. and interpret______

c. and interpret______

11. The cost of producing x teddy bears per day at the Cuddly Companion Company is calculated by their marketing staff to be given by the formula: C(x) = 500 + 50x – 0.0001x2.

a. Find the marginal cost function and use it to estimate how fast the cost is going up at a production

level of 200 teddy bears.

b. Find the average cost function, , and evaluate . What does this answer tell you?

c. At a production level of 200, is the average cost increasing or decreasing as more bears are produced?

12. Example 3, Page 202.

HW Page 205: 2, 5, 6, 9,10,11,13, 14, 15, 16, 21(Do not compare with actual cost on any of problems.)